Time-dependent screening in a two-dimensional electron gas

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236 M. Alducin et al. / Surface Science 559 (2004) 233–240 q ind ðrÞ ¼ Z Z 1 1 1 dqqJ 0 ðqrÞRe 1 ; 2p 0 2D ðq; 0Þ ð5Þ which correspond to the stationary expression for the density induced by an external charge density q ext ðrÞ ¼Z 1 dðrÞ. We checked that the results obtained by omitting the cosine term in Eq. (4) reproduce well those obtained with Eq. (5). Notice, however, that in order to achieve this agreement it was necessary to use the exact x p ðqÞ instead of the analytical approximations. 3. Results and discussion Fig. 1. Spectrum of excitations for (a) 2D electron gas (upper panel) and (b) 3D electron gas (lower panel). In both cases r s ¼ 2. The shadow areas indicate the ðq; xÞ-values at which the e-h pair excitations can take place. The curves x p ðqÞ indicate the plasmon excitations. In the 2D case however, the plasmon frequency behaves as x 2 p ’ q2 F q for small values of q and as x 2 p ’ q2 F q þ 3=4q2 F q2 for larger values of q: There are two distinctive features for the 2Dplasmon frequency as compared to the 3D one: x p is zero for q ¼ 0 and the q-dispersion is stronger. Let us make a final comment on the expression we derived for the induced density in Eq. (4). In the limit t !1, since the cosðxtÞ term integrates to zero, the x-integral can be analytically evaluated using the Kramers–Kroning relations. Hence, one gets In this section we show results for an external charge created in an electron gas with r s ¼ 2. Calculations were also carried out for different metallic densities (r s ¼ 3, 4). In general, we observe that for lower densities, the response of the medium to screen the external impurity is slower. Except for this fact, the comparative analysis between the 2D and the 3D electron gas presented in this section can be qualitatively generalized to those densities. In Fig. 2 we plot the electronic density n ind ¼ q ind induced at the position of the suddenly created charge as a function of time t for a 2D (Fig. 2(a), upper panel) and a 3D (Fig. 2(b), lower panel) electron gas. Since the spectral functions Imf1= RPA 2D;3D g consist of a continuum part (e-h pair) and a plasmon mode, we calculate the induced density as q ind ¼ q eh þ q pl , where q eh and q pl take into account the contribution to the screening process coming from e-h and plasmon excitation, respectively. Accordingly in Fig. 2, n ind (bold-solid line) is the sum of n eh (long-dashed line) and n pl (dotted line). The stationary value (t !1) is indicated by a thin-solid line. As it may be seen from the figure, n ind ð0; tÞ is dominated by the e-h part that rises quite fast within the first 2–3 a.u. of time. This behaviour is observed for both the 2D and the 3D medium. Nevertheless, an important difference arises between both cases. Whereas in a 2D electron gas n ind reaches the stationary value at t 8–9 a.u., in the 3D case n ind oscillates around its stationary value even for the longest time
M. Alducin et al. / Surface Science 559 (2004) 233–240 237 n ind (0, t) (a.u.) n ind (0, t) (a.u.) 0.3 0.2 0.1 0 0.08 0.06 0.04 0.02 (a) 2D 0 5 10 15 20 25 t (a.u.) (b) 3D 0 0 5 10 15 20 25 t (a.u.) Fig. 2. Time dependence of the electronic density n ind ð0; tÞ (bold-solid line) induced in a 2D electron gas (upper panel) and a 3D electron gas (lower panel) at the position of the suddenly created charge Z 1 ¼ 1. In both cases r s ¼ 2. The contribution to the induced density due to plasmon excitation n pl and to e-h pair excitations n eh are represented by dotted and by long-dashed lines, respectively. The steady-state value n ind ð0; t !1Þis indicated by a thin-solid line. shown in the figure. Comparison between Fig. 2(a) and (b) reveals that the origin of this discrepancy is, precisely, the different time dependence of n pl . As it was already pointed out by Canright for a 3D electron gas [9], the long-term transient is provided by the plasmon part. The existence of a quite welldefined frequency at which the spectral function takes the highest values gives rise to the oscillatory behaviour with frequency x p . The absence of this oscillatory term at long times in the 2D case is thus related to the special character of the 2D plasmon mode. The two features remarked in the previous section make difficult to find a unique characteristic frequency as in the 3D case. As a consequence, its contribution to the induced density at r ¼ 0 is very different from the 3D result, and does not show an oscillatory behaviour. In Fig. 3, we show the interpolated images of the radial densities, i.e., 2prn ind ðr; tÞ in 2D (Fig. 3(a)) and 4pr 2 n ind ðr; tÞ in 3D (Fig. 3(b)). The corresponding stationary limit is represented on the right part of each panel. We observe the Friedel oscillations appearing in both stationary limits that scale as [sinðq F rÞ=r] for r !1. The images provide a valuable information on the propagation of the response of the medium to an external perturbation. In view of these figures, the evolution in time of the screening process can be described in two steps: the switching on process or propagation of the shock-wave along the medium and the stabilization to the steady behaviour. We define the switching on process at a given distance as the interval going from the arrival of the first perturbation up to the instant at which the induced density reaches its highest value. This wave-front can be easily identified in each image as the first band growing upwards from the left-bottom corner of the figure. Considering this switching on process, we find similarities between the 2D and the 3D cases. The wave-front is characterized by strong variations of the induced density that achieves values much larger than those of its steady-state limit. The propagation velocity can be roughly estimated from the slope of the wave-front as v ð1 2Þv F (v F is the Fermi velocity in each case). Main differences appear in the transition to the stationary value. In the 2D case, the screening transients last roughly the passage of this wave-front. As a consequence, after a limited short-time interval since the arrival of the initial perturbation (Dt 8–10 a.u.), the steady-state value is basically reached. Furthermore, the transient regime does not show any oscillatory behaviour. Note that after 20 a.u. of time the stationary regime has been achieved for all the distances shown in the figure. On the contrary, in the 3D case the screening transients last for very long times. After the passage of the first wave-front the radial density continues oscillating around its steady-state value. In order to get information on the efficiency of the medium to screen the external charge, we also
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Time-dependent screening in a two-dimensional electron gas

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